On the List-Decodability of Random Self-Orthogonal Codes

TL;DR

This paper proves that random Euclidean self-orthogonal and symplectic dual-containing codes can achieve optimal list-decodability bounds, extending classical results to these specialized code classes and their quantum counterparts.

Contribution

It demonstrates that random Euclidean self-orthogonal and symplectic dual-containing codes achieve the same list-decodability bounds as general random codes, including quantum codes.

Findings

Self-orthogonal codes achieve the Gilbert-Varshamov bound in list decoding.

Symplectic dual-containing codes also reach the quantum Gilbert-Varshamov bound.

The counting argument is key to establishing these results.

Abstract

In 2011, Guruswami-H{\aa}stad-Kopparty \cite{Gru} showed that the list-decodability of random linear codes is as good as that of general random codes. In the present paper, we further strengthen the result by showing that the list-decodability of random {\it Euclidean self-orthogonal} codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. Specifically, we show that, for any fixed finite field $\F_q$, error fraction $\delta\in (0,1-1/q)$ satisfying $1-H_q(\delta)\le \frac12$ and small $\epsilon>0$, with high probability a random Euclidean self-orthogonal code over $\F_q$ of rate $1-H_q(\delta)-\epsilon$ is $(\delta, O(1/\epsilon))$-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding {\it symplectic dual-containing} codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.